For this equation x^(2) + x + 2
which you can't factor
but then you can use the quadratic formula
but than why is the answer no real solutions when I can use the quadratic formula?
did you mean the equation
x^2 + x + 2 = 0 ?
When you solve an equation of the form
ax^2 + bx + c =0 you are really just asking,
"where does the graph of the corresponding function
y = ax^2 + bx + c cross the x-axis"?
In your case the graph, which is a parabola, lies totally above the x-axis, thus no x-intercepts , and thus no real solution.
If the b^2 - 4ac part of the formula is zero, there will be only one solution.
If the b^2 - 4ac part of the formula is positive, there will be 2 different real solutions.
If the b^2 - 4ac part of the formula is negative, like in your case, there will be no real solution at all.
(there will be two imaginary solutions)
thank you
I forgot about having a negative for b^2 - 4ac was no real solutions
The equation x^2 + x + 2 cannot be factored easily, so you'll need to use the quadratic formula to find its solutions. The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions can be found using the formula:
x = (-b ± √(b^2 - 4ac))/(2a)
For the equation x^2 + x + 2, comparing it to the general form ax^2 + bx + c = 0, we have:
a = 1, b = 1, and c = 2.
Plugging these values into the quadratic formula, we get:
x = (-1 ± √(1^2 - 4(1)(2)))/(2(1))
Simplifying further:
x = (-1 ± √(-7))/2
Here's where the issue arises. The expression inside the square root (√(-7)) represents a negative number. In the real number system, the square root of a negative number is not defined. Therefore, there are no real solutions to the equation x^2 + x + 2.
However, it's important to note that using the quadratic formula still gives you an answer. In this case, the answer involves complex numbers. Complex numbers include both real and imaginary components and are represented as a + bi, where a is the real part and bi is the imaginary part.
So, the solutions to the equation x^2 + x + 2 are complex numbers, specifically:
x = (-1 + i√7)/2 and x = (-1 - i√7)/2, where i is the imaginary unit (√(-1)).